Abstract
In this paper, we discuss under which conditions cyclic essential extensions of simple modules over a differential operator ring R[θ;δ] are Artinian. In particular, we study the case when R is either δ-simple or δ-primitive. Furthermore, we obtain important results when R is an affine algebra of Kull dimension 2. As an application, we characterize the differential operator rings for which cyclic essential extensions of simple modules are Artinian.
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